Best Known (30, 71, s)-Nets in Base 27
(30, 71, 140)-Net over F27 — Constructive and digital
Digital (30, 71, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 24, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (6, 47, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (4, 24, 64)-net over F27, using
(30, 71, 172)-Net in Base 27 — Constructive
(30, 71, 172)-net in base 27, using
- 21 times m-reduction [i] based on (30, 92, 172)-net in base 27, using
- base change [i] based on digital (7, 69, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 69, 172)-net over F81, using
(30, 71, 211)-Net over F27 — Digital
Digital (30, 71, 211)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2771, 211, F27, 2, 41) (dual of [(211, 2), 351, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2771, 213, F27, 2, 41) (dual of [(213, 2), 355, 42]-NRT-code), using
- construction X applied to AG(2;F,372P) ⊂ AG(2;F,379P) [i] based on
- linear OOA(2765, 207, F27, 2, 41) (dual of [(207, 2), 349, 42]-NRT-code), using algebraic-geometric NRT-code AG(2;F,372P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208, using
- linear OOA(2758, 207, F27, 2, 34) (dual of [(207, 2), 356, 35]-NRT-code), using algebraic-geometric NRT-code AG(2;F,379P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208 (see above)
- linear OOA(276, 6, F27, 2, 6) (dual of [(6, 2), 6, 7]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(276, 27, F27, 2, 6) (dual of [(27, 2), 48, 7]-NRT-code), using
- Reed–Solomon NRT-code RS(2;48,27) [i]
- discarding factors / shortening the dual code based on linear OOA(276, 27, F27, 2, 6) (dual of [(27, 2), 48, 7]-NRT-code), using
- construction X applied to AG(2;F,372P) ⊂ AG(2;F,379P) [i] based on
- discarding factors / shortening the dual code based on linear OOA(2771, 213, F27, 2, 41) (dual of [(213, 2), 355, 42]-NRT-code), using
(30, 71, 298)-Net in Base 27
(30, 71, 298)-net in base 27, using
- 1 times m-reduction [i] based on (30, 72, 298)-net in base 27, using
- base change [i] based on digital (12, 54, 298)-net over F81, using
- net from sequence [i] based on digital (12, 297)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 12 and N(F) ≥ 298, using
- net from sequence [i] based on digital (12, 297)-sequence over F81, using
- base change [i] based on digital (12, 54, 298)-net over F81, using
(30, 71, 32656)-Net in Base 27 — Upper bound on s
There is no (30, 71, 32657)-net in base 27, because
- 1 times m-reduction [i] would yield (30, 70, 32657)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 15689 127124 211954 663144 479402 082001 294318 584720 739684 921451 298881 654319 824660 003500 839585 689769 775153 > 2770 [i]